Slope-Intercept Form (Part 1)

Welcome! In this lesson, I’ll derive the equation to a slanted line, which looks like this:

Slope Intercept Form

Given that, the line makes a certain angle (θ) with the X axis, and passes through a point on the Y axis which is at a certain distance ‘c’ from the origin.

But do we really need both the angle and ‘c’? What if only the angle were given? What if only ‘c’ was given?

If only one of these were given, then there wouldn’t have been a unique line possible. Look at the figures below.

So we’ll need both the angle and ‘c’, for there to be a unique line. In general, as we’ll see later, we’ll require two given conditions to uniquely determine a line on the XY plane.

By the way this ‘c’ is the y-intercept, which will be positive or negative depending upon whether the point (at which the line intersects the Y axis) lies above or below the origin. (It can also be zero, of course)

And the tangent of the angle made by the line (i.e. tanθ) is known as the slope of the line. The convention is to measure the angle in the anticlockwise direction with the X-axis. (which will make the angle lie in the range [0, π) )

Now, we have a unique line, whose slope and y-intercept is given. On to its equation !

Lets take general point P(x, y) on the line, and try to find a relation which will always hold true (the definition!)

Too much of a mess there.. PA perpendicular to the X axis.. and CB parallel.. don’t run off yet !